What is the meaning of "wearing one's learning lightly"?

Solution 1:

Wear one's learning lightly

This means to be knowledgeable but not boast about it or seek undue privilege from it. It means to behave in a modest and understated manner.

They knew what they were about

This is a slightly old-fashioned phrase, which in modern times is usually expressed

They knew what they were talking about

It means their remarks were knowledgeable and insightful. In contrast, the phrase

He doesn't know what he is talking about

is applied to people who make statements that sound impressive, but are incorrect (in the sense of being confused rather than being intentionally deceptive).

To answer your question, "they" represents the "critics of the major papers", such as Shaw and Newman.[thanks to James Random for highlighting in comments below]

What the writer is saying is that that generation of journalists was very knowledgeable and took their work seriously. Even those who were modest and did not make a big noise about their own high level of learning, actually did know a lot about the subject matter, and their opinions were well-reasoned.

In the closing quote, Newman is pointing out that journalism is a very demanding discipline, and many writers do not have the aptitude for it. Moreover he suggests that some writers who cannot achieve at this high level pretend that journalism is a lower form of literature, to enhance their own self-esteem. His final twist of the knife is to suggest that more people read a journalists' writing than a non-journalists'.